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Talk

Hyperbolic Monge-Ampere equations and applications to thin sheet elasticity

  • Shankar Venkataramani (The University of Arizona)
A3 01 (Sophus-Lie room)

Abstract

I will talk about some geometric questions that arise in the study of soft/thin objects with negative curvature. I will discuss some recent results on existence/non-existence of solutions to hyperbolic Monge-Ampere equations with various degrees of regularity, with an emphasis on numerical methods for constructing "rough" solutions. I will then discuss some applications of our results to (i) the occurrence of "geometric" defects that are invisible to the energy, but play a crucial role in determining the global morphology, (ii) a generalization of the Sine-Gordon equation to describe "rough" hyperbolic surfaces with constant negative curvature, and (iii) the important role of regularity in quantitative versions of the Hilbert-Efimov theorem on the nonexistence of C^2 isometric immersions of the Hyperbolic plane into R^3, and (iv) studying the mechanics of leaves, flowers, and sea-slugs.

This is joint work with Toby Shearman and Ken Yamamoto.

Upcoming Events of this Seminar

  • Dienstag, 11.03.25 tba with Jianfeng Lu
  • Dienstag, 11.03.25 tba with Steffen Börm
  • Dienstag, 06.05.25 tba with Ilya Chevyrev
  • Dienstag, 03.06.25 tba with Mate Gerencser